The rarefied flow of monatomic gas between an inner rotating cylinder and an outer stationary cylinder is simulated by the Monte Carlo direct simulation method. The flow is assumed to be axisymmetrical. A new idea to calculate the axisymmetrical flow is proposed. Simulations are performed for Knudsen numbers in the range 0.004<Kn<0.04 at a fixed rotation speed. The calculated torque coefficients show a very good agreement with experimental data not only for vortex free but also for vortical flow regimes. Once the Taylor vortices have developed, they cause a change of the circumferential velocity profile up to 35%. This is the first numerical simulation of Taylor vortices via solutions of the Boltzmann equation using the molecular approach.